Related papers: Optimal Quantization on Spherical Surfaces: Contin…
The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a…
We show that the theory of varifolds can be suitably enriched to open the way to applications in the field of discrete and computational geometry. Using appropriate regularizations of the mass and of the first variation of a varifold we…
Density estimation is an important technique for characterizing distributions given observations. Much existing research on density estimation has focused on cases wherein the data lies in a Euclidean space. However, some kinds of data are…
Many computer vision challenges require continuous outputs, but tend to be solved by discrete classification. The reason is classification's natural containment within a probability $n$-simplex, as defined by the popular softmax activation…
We propose a group sparse optimization model for inpainting of a square-integrable isotropic random field on the unit sphere, where the field is represented by spherical harmonics with random complex coefficients. In the proposed…
As the most common representation for 3D shapes, mesh is often stored discretely with arrays of vertices and faces. However, 3D shapes in the real world are presented continuously. In this paper, we propose to learn a continuous…
We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation…
Spherical $t$-designs on $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$ provide $N$ nodes for an equal weight numerical integration rule which is exact for all spherical polynomials of degree at most $t$. This paper considers the generation of…
In this work, we extend the classical framework of quantization for Borel probability measures defined on normed spaces $\mathbb{R}^k$ by introducing and analyzing the notions of the $n$th constrained quantization error, constrained…
The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability…
The work develops further the theory of the following inversion problem, which plays the central role in the rapidly developing area of thermoacoustic tomography and has intimate connections with PDEs and integral geometry: {\it Reconstruct…
For a distribution function $F$ on $\mathbb{R}^d$ and a point $q\in \mathbb{R}^d$, the \emph{spherical depth} $\SphD(q;F)$ is defined to be the probability that a point $q$ is contained inside a random closed hyper-ball obtained from a pair…
We investigate the optimal configurations of n points on the unit sphere for a class of potential functions. In particular, we characterize these optimal configurations in terms of their approximation properties within frame theory.…
The basic elements of the geometric approach to a consistent quantization formalism are summarized, with reference to the methods of the old quantum mechanics and the induced representations theory of Lie groups. A possible relationship…
Let $\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$ denote the set of $C^1$ regular curves in the $2$-sphere $\mathbb{S}^2$ that start and end at given points with the corresponding Frenet frames $\boldsymbol{P}$ and…
Many applications of geometry modeling and computer graphics necessite accurate curvature estimations of curves on the plane or on manifolds. In this paper, we define the notion of the discrete geodesic curvature of a geodesic polygon on a…
Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and…
We consider the problem of computing the topology and describing the geometry of a parametric curve in $\mathbb{R}^n$. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space.…
We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is…
We introduce a novel family of projected distributions on the circle and the sphere, namely the circular and spherical projected Cauchy distributions, as promising alternatives for modelling circular and spherical data. The circular…